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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Between reduced powers and ultrapowers, II.
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by Ilijas Farah and Saharon Shelah PDF
Trans. Amer. Math. Soc. 375 (2022), 9007-9034 Request permission


We prove that, consistently with ZFC, no ultraproduct of countably infinite (or separable metric, non-compact) structures is isomorphic to a reduced product of countable (or separable metric) structures associated to the Fréchet filter. Since such structures are countably saturated, the Continuum Hypothesis implies that they are isomorphic when elementarily equivalent.
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Additional Information
  • Ilijas Farah
  • Affiliation: Department of Mathematics and Statistics, York University, 4700 Keele Street, North York, Ontario, M3J 1P3, Canada; and Matematički Institut SANU, Kneza Mihaila 36, Belgrade 11001, Serbia
  • MR Author ID: 350129
  • ORCID: 0000-0001-7703-6931
  • Email:
  • Saharon Shelah
  • Affiliation: The Hebrew University of Jerusalem, Einstein Institute of Mathematics, Edmond J. Safra Campus, Givat Ram, Jerusalem 919041, Israel and Department of Mathematics, Hill Center-Busch Campus, Rutgers, The State University of New Jersey, 110 Frelinghuysen Road, Piscataway, New Jersey 08854-8019
  • MR Author ID: 160185
  • ORCID: 0000-0003-0462-3152
  • Email:
  • Received by editor(s): November 14, 2020
  • Received by editor(s) in revised form: August 12, 2021, June 4, 2022, and July 6, 2022
  • Published electronically: September 29, 2022
  • Additional Notes: The first author’s research was partially supported by NSERC. The second author’s research was partially supported by Israel Science Foundation (ISF) grant no: 1838/19 and by NSF grant no: DMS 1833363. No. 1202 on the second author’s list of publications.
  • © Copyright 2022 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 375 (2022), 9007-9034
  • MSC (2020): Primary 03C20, 03C98, 03E50, 03E35
  • DOI: